Fourier series question

Question

How do we know that a Fourier series expansion does exist for a given function $f(x)$? I mean, if $f(x)=x$ and we suppose that $x=a_1\sin(x)+a_2\sin(2x)$ with $-\pi\leq x \leq \pi$ the Fourier coefficients $a_1$ and $a_2$ still being the same as if we suppose that $x=a_1\sin(x)+a_2\sin(2x)+...$ So, why there are infinitely many sine or cosine terms?

Answer 1

I think your question has two parts. The first is existence for a given $f$. A Fourier series exists for a given $f(x)$, $x \in [-\pi,\pi)$ when

$$\int_{-\pi}^{\pi} dx \, |f(x)|^2 \lt \infty$$

The second part I think asks how do we know that the FS coefficients do not change when we add more harmonics to the series. This is because the coefficients do not depend on the number of terms in the sum; they simply depend on the particular harmonic and the function, i.e.,

$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} dx \, f(x) \, \sin{n x}$$